Problems in Geometry
(5)
1.
Given a circle
γ,
consider points
P, C, D
∈
γ
and let [
A, B
] be a diameter of
γ .
Prove
that the Simson line of
P
with respect to the triangle
ACD
is perpendicular to the Simson
line of
P
with respect to the triangle
BCD.
1
2.
Given a triangle
ABC
with circumcenter
O,
let
A
0
, B
0
, C
0
be the respective midpoints
of the line segments [
B, C
]
,
[
C, A
]
,
[
A, B
]
.
Let
E
be the foot of the perpendicular from
B
on
CA
and
U, W
be the feet of perpendiculars from
E
on
B
0
C
0
, A
0
B
0
respectively. Prove
that
WU
bisects [
B, E
] and is
2
parallel to
OB .
3.
Given a triangle
ABC,
let the parallel to
BC
through
A
meet the circumcircle of
ABC
in
L
for a second time.
(A) Prove that the Simson line of
L
is parallel to
OA
where
O
is the circumcenter of
ABC.
(B) Let [
A, V
] and [
L, U
] be diameters of the circumcircle of
ABC.
Describe the Simson
lines of
U
and
V.
4.
Consider a triangle
ABC
with circumcenter
O .
For each
X
on the circumcircle of
ABC
let
l
X
denote the Simson line of
X
with respect to
ABC.
(A) Prove that in order for
l
X
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 Spring '11
 cemtezer
 Geometry, Regular polygon, Simson line

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